Homogeneous Linear Equation System

A system of linear equations is said to be homogeneous if all constant terms are zero, that is, the system has the form:

Each system of homogeneous linear equations is a consistent system, because

x₁=0, x₂=0, ..., x_n=0

always a solution. The solution is called trivial solution. If there is another solution, it is called a nontrivial solution,

Because a system of homogeneous linear equations is always consistent, there will be one solution or infinite number of solutions. Because one of these solutions is a trivial solution, we can make the following statement:

For a system of homogeneous linear equations, then exactly one of the following statements is true.

1. The system only has trivial solutions.
2. The system has an infinite number of non-trivial solutions.

There is one case where the homogeneous system is certain to have a non-trivial solution, that is, if the system involves more unknown numbers than many equations. To see why this is so, review the following example of four equations with five unknown numbers.

The enlarged matrix for the system is

By reducing this matrix to a reduced line echelon, we get,

The system of equations corresponding to this matrix is

With the completion of the main variables it will produce

Then the solution will be given by

x₁=-s-t, x₂=s, x₃=-t, x₄=0, x₅=t

Note that trivial solving can be obtained if s = t = 0

The example above illustrates two important things about how to solve a system of homogeneous linear equations, i.e.

1. None of the three basics of the row operation can change the last zero column in the enlarged matrix, so that the system of equations corresponding to the reduced echelon form of the enlarged matrix must also be a homogeneous system.
2. Depending on whether the reduced row echelon form of the enlarged matrix has a zero row, the number of equations in the reduced system is equal to or smaller than the number of equations in the original system (compare systems (*) and systems (**) in the example above.

So the homogeneous system given has m equations with n unknown numbers and m < n, and if r nonzero rows are in the form of a reduced line echelon of an enlarged matrix, we will have r < n, so the system of equations corresponding to the shape of the reduced line echelon of the enlarged matrix will look like:

Where         are the main variables and ∑ () states the number that involves remaining variable n-r. By completing for the main variables will be obtained.

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